Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

The space of states is acted on 2-linearly by pull-push through spanshist↙↘confconf,
\array{
&&
\mathrm{hist}
\\
& \swarrow
&&& \searrow
\\
\mathrm{conf}
&&&&&
\mathrm{conf}
}\,,
which may encode operation like time evolution or gauge transformations like T-duality.

In a chosen 2-basis for VV, which is an algebra, 2-states appear as modules and 2-linear maps appear as bimodules.

The former fact harmonizes with the term “gerbe module” used for D-branes. In that sense, these bimodules could be addressed as bi-branes.

Like an ordinary brane – at least in its geometric incarnation as a subspace with Chan-Paton bundle on it – is a submanifold of target space over wich the Kalb-Ramond field strength (the curvature of the gerbe) trivializes, a bi-brane is defined to be a submanifold of two different target spaces, over which the difference of two KR-fields trivializes.

While only very briefly toughed upon in the above paper, this is the familiar central structure of interest in topological T-duality, in which case the bi-brane bundle is the Poincaré-line bundle.

In fact, the condition on the KR fields now given for bi-branes is known in topological T-duality, as for instance discussed on p. 5 of

Some Related Entries

11 Comments & 5 Trackbacks

Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

This is interesting and I feel somehow related to some work I’m doing on orthogonal hermite polynomials in 2-variables and pointwise fourier integral transforms and diagonalization via projective integral transforms via the Gram-Schmidt process. All my work is done via hypergeometric functions, generating functions, and quite a bit of analytic number theory. The nice thing is that every single step of the way I am only working with integer sequences and renormalized power series expansions. So little ole me has been able to do this stuff concretey without all this abstract nonsense. So, why is all this abstract stuff so popular when number theory is so much more concrete and applicable?

two things and 2-things

[…] polynomials in 2-variables […]

I am a little worried that what you mean are polynomials in two variables, i.e. elements in some K[x,y]K[x,y].

Is that so? In that case I’d unfortunately have to disappoint you: when we say “2-vectors” we don’t just mean two vectors (as in: “a pair of vectors”), but a notion of vector that is to an ordinary vector like a category is to a set.

On the other hand, if you really do mean polynomials in categorified variables, then I’d love to know more details on what exactly it is you are considering!

I should maybe remark that, after all, there is often a simple special case of a 2-something which does appear more or less as nothing but two somethings.

For instance, the 2-vector spaces V\mathbf{V} described and studied by John Baez and Alissa Crans in HDA IV are , in particular, two vector spaces V0V_0 and V1V_1V={V0,V1},
\mathbf{V} = \{V_0, V_1\}
\,,
namely a vector space, V0V_0, of objects and a vector space, V0⊕V1V_0 \oplus V_1of morphisms.

What makes the two vector spaces a 2-vector space is extra structure on this, given by maps
s,t:V0⊕V1→V0
s,t : V_0 \oplus V_1 \to V_0
and
i:V0→V0⊕V1
i : V_0 \to V_0 \oplus V_1
and
∘:(V0⊕V1)[2]→V0⊕V1
\circ : (V_0\oplus V_1)^{[2]}
\to V_0 \oplus V_1
satisfying the usual laws which say that ∘\circ is the composition of morphism which have source ss and target tt.

Again, the reason for this is that a strict 2-group, like a Baez-Crans 2-vector space, is an internal category: BC 2-vector spaces are categories internal to ordinary vector spaces, while strict 2-groups are categories internal to ordinary groups.

Since a category consists, in particular, of two objects, namely an object of objects and an object of morphisms, this gives rise to the phenomenon that 2-things often look like two things .

Sometimes (rarely, but sometimes), it indeed happens that people first study a theory of two things (like of two variables) with extra structure and properties around, and only later realize (or somebody else does, usually ;-) that what they are really studying is a 2-thing.

A nice example of this are 2-class functions and their categorical interpretation.

Like a class function on a group is a function g↦f(g)g \mapsto f(g) of a single variable that is invariant under conjugation, an nn-class function(g1,g2,⋯,gn)↦f(g1,g2,⋯,gn)
(g_1,g_2,\cdots, g_n)
\mapsto
f(g_1,g_2,\cdots, g_n)
is a function of nn-tuples of (pairwise commuting) group elements, that is invariant under simultaneous conjugation of these group elements.

So an nn-class function is, in particular, a funcion of nn-variables. People found that to be of use in certain contexts.

Then along came Ganter and Kaparanov and showed that like class functions come from traces of representations of ordinary groups, 2-class functions can be seen as coming from 2-traces of (2-)groups!

I think there are more examples of this kind. In fact, here we enjoy going through standard literature and trying to spot conglomerates of structures that secretly arrange themselves into single, unified nn-categorical structures.

A more intricate example of this, for instance, is the local data for connections on gerbes. This comes in large parts of the literature as a vast array of various pp-forms with values in various groups and Lie algebras. But staring at this structure for a while reveals that all this mess are nothing but the components of one single morphism between higher-groupoids.

There are more examples like that, but I think I’ll stop here.

If you knew all this and really do mean that you are working with categorified variables then I apologize for wasting your time with this lengthy reply here, hoping that maybe others find it helpful.

Re: n-class functions and n-traces

Since I’ve been sneakily co-opted into this conversation, let me add my two cents worth.

David asked :

Is the n-class function story expected to continue? Are 3-class functions expected to come from 3-traces of (3-)groups?

It seems that the pattern should indeed continue - that’s the way I understand it, at least. One thing though - I understand it as n-class function coming from n-representations of groups, not just from a n-group. I guess that’s what you guys meant as well; I mean what is a “trace” of a group element?

Lets recall the pattern. You can take the character of an ordinary representation of a groupoid GG to get a class function, i.e. a section of the inertia groupoid ΛG\Lambda G:

(1)χ:Rep(G)→ΓΛG.
\chi : Rep(G) \rightarrow \Gamma_{\Lambda G}.

In particular, if GG is a group, taking the character of a representation of ΛG\Lambda G gives us a section of Λ2G\Lambda^2 G:

Taking the character again will give us a section of Λ3G\Lambda^3 G, i.e. a 3-class function .

By the way, 2Rep(ΛG)2Rep (\Lambda G) is a really cool 2-category to consider. Just like Rep(ΛG)Rep(\Lambda G) is a braided monoidal category, 2Rep(ΛG)2Rep(\Lambda G) is a braided monoidal 2-category with duals! Anyone want to help me make this rigourous?

The generating function for this transform can be found.. derived it heuristically and its unkown to me whether there is a systematic way to do it.

In any case, after the fourier transform the set is no longer orthogonal or normal, so it can be renormalzed again, and then the Gram-Schmidt orthogonalization process can be applied by taking the complex inner products using the complex conjugates and sucessively projecting the 0th vector onto the 1st, then the 1st projection onto the 2nd, 2nd onto 3rd, etc.

This process is burdensome but I wrote some maple code to do it fairly quickly and then work on deriviving generating functions again heuristically. I noticed that the imaginary component of the even fourier transforms =0 forall n, and the real part of the odd dimensional transforms =0 forall n, Based on this I decmoposed the transform into even and odd componets, and apply the Gram-schmidt process to each piece and then divide the dimension by 2 and sum to get an infinite seqeuence of complex orthonormal distribution functions which rapidly approach 0, and t always stays real. It’s the space variable that is complexified.

I believe the things im studying are the ‘Bochern-Riesz means’ related to the Riesz-Fischer theorem.

I have the great book ‘Intruduction to Geometric Probability’ by Klain and Rota where I’ve found their presentation of ‘valuations on polyconvex lattices’ and the Euler characteristic very nice. Specfically, Section 8.2 on ‘Even and Odd Valuations’, which I believe is exactly what I’ve found with the real/even odd/imaginary decomposition of the Fourier transform I’ve found. The hypergeometric function I listed above was studied by Wiener (although not in this hypergeometric form) and he called it the ‘polynomial chaos’, and the closely related ‘Differential Space’.

Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

I think this is mostly off-topic, aside from the word “gerbe”: some time ago you had an interesting post on the String Coffee Table about the n-cubed scaling in the theory on a stack of 5-branes. In your opinion, is the n-cubed scaling now understood, from the perspective of work on gerbes or 2-gauge theories or anything else along these lines? Has any of this shed light on what the right degrees of freedom are to see the counting explicitly? Any links would be much appreciated.

Read the post Report from "Workshop on Higher Gauge Theory"Weblog: The n-Category CaféExcerpt: Report-back on a little symposium titled "Higher Gauge Theory" (but concerned just with abelian gerbes) that took place at the AEI in Golm.Tracked: May 9, 2007 11:50 AM

Read the post Wilson Loop Defects on the StringWeblog: The n-Category CaféExcerpt: On Alekseev and Monnier's work on quantizing Wilson loop observables for the WZW model.Tracked: August 23, 2007 5:01 PM

Read the post Planar Algebras, TFTs with DefectsWeblog: The n-Category CaféExcerpt: I am in Vienna at the ESi attending a few days of the program Operator algebras and CFT. This morning we had a nice talk by Dietmar Bisch on Dietmar Bisch, Paramita Das, Shamindra Kumar Ghosh The planar algebra...Tracked: September 11, 2008 3:51 PM